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27 Oct 2008 Masaki Kashiwara and Algebraic Analysis Masaki Kashiwara and Algebraic Analysis Pierre Schapira Abstract This paper is based on a talk given in honor of Masaki Kashiwara’s 60th birthday, Kyoto, June 27, 2007. It is a brief overview of his main contributions in the domain of microlocal and algebraic analysis. It is a great honor to present here some aspects of the work of Masaki Kashiwara. Recall that Masaki’s work covers many fields of mathematics, algebraic and microlocal analysis of course, but also representation theory, Hodge the- ory, integrable systems, quantum groups and so on. Also recall that Masaki had many collaborators, among whom Daniel Barlet, Jean-Luc Brylinski, Et- surio Date, Ryoji Hotta, Michio Jimbo, Seok-Jin Kang, Takahiro Kawai, Tet- suji Miwa, Kiyosato Okamoto, Toshio Oshima, Mikio Sato, myself, Toshiyuki Tanisaki and Mich`ele Vergne. In each of the domain he approached, Masaki has given essential contri- butions and made important discoveries, such as, for example, the existence of crystal bases in quantum groups. But in this talk, I will restrict myself to describe some part of his work related to microlocal and algebraic analysis. The story begins long ago, in the early sixties, when Mikio Sato created a new branch of mathematics now called “Algebraic Analysis”. In 1959/60, arXiv:0810.4875v1 [math.HO] 27 Oct 2008 M. Sato published two papers on hyperfunction theory [24] and then devel- oped his vision of analysis and linear partial differential equations in a series of lectures at Tokyo University (see [1]). If M is a real analytic manifold and X is a complexification of M, hyperfunctions on M are cohomology classes supported by M of the sheaf X of holomorphic functions on X. It is difficult O to realize now how Sato’s point of view was revolutionary at that time. Sato’s hyperfunctions are constructed using tools from sheaf theory and complex analysis, when people were totally addicted to functional analysis, and when the separation between real and complex analysis was very strong. 1 Figure 1: Mikio Sato around 1972 and with Masaki Kashiwara more recently Then came Kashiwara’s thesis, dated December 1970 (of course written in Japanese, but translated in English and published by the French Mathe- matical Society [7]) in which he settles the foundations of analytic -module theory and obtains almost all basic results of the theory (compare withD [13]). With -module theory (also constructed independently in the algebraic set- tings byD J. Bernstein [3]), one finally has the tools to treat general systems of linear partial differential equations, as opposed to one equation with one unknown, or to some very particular overdetermined systems. In particular, Kashiwara succeeds in formulating (and solving, but the difficult problem is the functorial formulation) the Cauchy problem for -modules, obtaining D what is now called the Cauchy-Kowalesky-Kashiwara theorem. After the first revolution of hyperfunction theory, Sato made a second one, ten years later, by creating microlocal analysis, a way to analyse objects of a manifold X in the cotangent bundle T ∗X. With Kashiwara and Kawai, they wrote a long paper [25], quoted everywhere as SKK, whose influence has been considerable during the whole seventies among the analysts (and not only the analysts), although very few of them even tried to read the paper. The SKK paper contains Sato’s construction of the sheaf M of mi- C crofunctions, and as a byproduct, the definition of the wave front set. This is essentially what the analysts, led by H¨ormander, remember of this theory (see [6]). But, to my opinion, this is certainly not the only key point of 2 the SKK paper. Another essential fact is that all constructions are made functorially. For example, microfunctions are obtained by first constructing the microlocalization functor µM , and then applying it to the sheaf X of holomorphic functions on a complex manifold X. When you take forOM a real analytic manifold of whom X is a complexification, you get the sheaf M ∗ C (living on TM X, the cornormal bundle to M in X), but if you replace the embedding M ֒ X by the diagonal embedding ∆ ֒ X X, then you get → ∗ → ×∗ the sheaf of microdifferential operators (on T∆(X X) T X) whose theory was developed by Kashiwara and Kawai. With this× approach,≃ you can adapt the six Grothendieck operations to Analysis and obtain a completely new point of view to classical problems (e.g. the Fourier-Sato transformation). Moreover, the SKK paper contains at least two fundamental and ex- tremely deep results, first the involutivity of characteristics, second the struc- ture of systems of microdifferential equations at the generic points of the characteristic variety. More precisely, let X be a complex manifold and let X be the sheaf of rings of microdifferential operators (a kind of localization E of the sheaf X of differential operators). A microdifferential system on D ∗ M an open subset U of T X is a coherent X U -module. Then E | the support char( ) of , also called its characteristic variety, is a • M M closed complex analytic involutive (that is, co-isotropic) subset of U. Of course, the involutivity theorem has a longer history, including the previous work of Guillemin-Quillen-Sternberg [5], and culminating with the purely algebraic proof of Gabber [4]. At generic points of char( ), (after using complex quantized contact • M transformations and infinite order microdifferential operators) is isomorphic to a partial de Rham system: M ∂xi u =0, (i =1,...,p). In the real case, is isomorphic to a mixture of de Rham, Dolbeault M and Hans Lewy systems: ∂x u =0, (i =1,...,p) i (∂y + √ 1∂y )u =0, (j =1,...,q) j − j+1 (∂tk + √ 1tk∂tk+1 )u =0, (k =1,...,r). − From 1970 to 1980, Kashiwara solved almost all fundamental questions of -module theory, proving in particular the rationality of the zeroes of D 3 Figure 2: Masaki Kashiwara, Teresa Monteiro Fernandes and myself around 1975 b-functions [10] and also stating and solving almost all questions related to regular holonomic modules, in particular the Riemann-Hilbert problem. Let us give some details on this part of Kashiwara’s work. In 1975, he proved that the complex F = R om D( , X ) of holomorphic solutions of a holonomic -module has constructibleH M O cohomology and satisfies proper- D M ties which are now translated by saying that F is perverse [9]. Moreover, two years before [8], in 1973, he calculated the local Euler-Poincar´eindex of F using the characteristic cycle associated to and in fact, defining first what M is now called the local Euler obstruction, or equivalently, the intersection of Lagrangian cycles. In 1977 he gave a precise statement of what should be the Riemann-Hilbert correspondence (see [23, p. 287]), the difficulty being to de- fine a suitable class of holonomic -modules, the so-called regular holonomic modules, what he does in the microlocalD setting with Kawai [15] (after re- lated work with Oshima [16]). Then, in 1979, he announces at the 1979/1980 Seminar of Ecole Polytechnique [11] the theorem, giving with some details 4 the main steps of the proof. (1975) b op b C Dhol( X ) / DC−c( X ) DO kkk ∼ kkk (1977) kkk kkk ? kk (1979−80) b opu D ( X ) holreg D Unfortunately, Masaki did not publish the whole proof before 1984 [12] and some people tried to make his result their own. As everyone knows, if the platonic world of Mathematics is pure and rigorous, these qualities definitely do not apply to the world of mathematicians. Of course, Kashiwara did a lot of other things during this period 1970/80, in particular in the theory of microdifferential equations, but he did not always take the time to publish his results. I remember that I had once in 1978 at Oberwolfach the opportunity to explain to H¨ormander the so-called “watermelon cut theorem” and you can now find it in [6, Th. 9.6.6]. This beautiful theorem asserts in particular that if a hyperfunction u is supported by a half space f 0, then the analytic wave front set of u above the ≥ boundary f = 0 is invariant by the Hamiltonian vector field Hf . After that, essentially from 1980 to 1990, came another period in which I am more involved. Indeed, we developed together the microlocal theory of sheaves (see [18]). To a sheaf F (not necessarily constructible) on a real manifold M, we asso- ciate a closed conic subset SS(F )1 of the cotangent bundle, the microsupport of F , which describes the directions of non propagation of F . The idea of microsupport emerged when, on one side, Masaki noticed that it was possi- ble to recover the characteristic variety of a holonomic -module from the knowledge of the complex of its holomorphic solutionsD by using the van- ishing cycle functor, and when, on my side, I was lead to this notion by remarking that our previous results on propagation for hyperbolic systems was of purely geometrical nature and had almost nothing to do with partial differential equations. One of the main result of the theory asserts that the microsupport of a sheaf F is an involutive subset of the cotangent bundle, but now we are working on real manifolds. In case one works on a complex manifold and F is the sheaf of solutions of a coherent -module , the microsupport of F D M 1SS(F ) stands for singular support. 5 Figure 3: Masaki Kashiwara and myself around 1980 coincides with the characteristic variety of . This gives an alternative proof M of the involutivity of characteristics of -modules.
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